[Paper Review] Selective Reverse PAC Coding for Sphere Decoding

Xinyi Gu, Mohammad Rowshan+1 more|arXiv (Cornell University)|Dec 1, 2022

Error Correcting Code Techniques4 citations

TL;DR

This paper proposes selective reverse PAC (SR-PAC) coding to enable sphere decoding of polar codes by applying convolutional precoding in reverse order, reducing the error coefficient (number of minimum-weight codewords) without degrading the minimum distance. The method achieves 0.2–0.6 dB SNR gain in block error rate, especially for high-rate codes and high SNR regimes, by preserving the code's minimum distance while minimizing error-prone codewords.

ABSTRACT

Convolutional precoding in polarization-adjusted convolutional (PAC) codes can reduce the number of minimum weight codewords (a.k.a error coefficient) of polar codes. This can result in improving the error correction performance of (near) maximum likelihood (ML) decoders such as sequential decoders and sphere decoders. However, PAC codes cannot be decoded by sphere decoding. The reason is twofold: 1) Sphere decoding of polar codes is performed from the last bit - due to the lower rectangular shape of the polar transform. Whereas the shape of PAC codes generator matrix is no longer triangular. 2) One may modify the precoding matrix to get a lower-triangular shape. However, this may reduce the minimum distance of the code due to the formation of unwanted cosets. This work proposes a selective convolutional precoding scheme with transposed precoding matrix to reduce the error coefficient while avoiding the reduction in the minimum distance. The numerical results show the improvement of block error rate by 0.2-0.6 dB, depending on the code rate, in medium and high SNR regimes.

Motivation & Objective

• To enable sphere decoding of polar codes by overcoming the structural mismatch caused by non-triangular generator matrices in PAC codes.
• To reduce the error coefficient (number of minimum-weight codewords) in polar codes to improve block error rate under maximum-likelihood decoding.
• To preserve the minimum distance of the original polar code while minimizing error coefficient through reverse precoding.
• To design a selective precoding scheme using transposed precoding matrices that avoids unintended coset formation.
• To demonstrate performance gains in high-rate and high-SNR regimes using sphere decoding with reduced search complexity.

Proposed method

• Proposes a selective reverse PAC (SR-PAC) coding scheme that applies convolutional precoding in reverse order to maintain a lower-triangular structure suitable for sphere decoding.
• Uses a transposed precoding matrix to ensure the generator matrix remains lower-triangular, enabling efficient tree search in sphere decoding.
• Analyzes the impact of convolutional polynomials on the formation of minimum-weight codewords in cosets, particularly focusing on coset leaders with minimal weight.
• Employs a systematic design where only specific information bits are precoded, avoiding full precoding that could reduce minimum distance.
• Introduces a criterion to select precoding polynomials that maintain the minimum distance by ensuring coset leaders have sufficient weight (e.g., w(g_i) = 16 for w_min = 16).
• Validates the method through numerical evaluation of block error rate (BLER) under sphere decoding, comparing against polar codes and SCL decoding.

Experimental results

Research questions

• RQ1 Can reverse precoding in PAC codes preserve the minimum distance while reducing the error coefficient for sphere decoding?
• RQ2 How does the choice of convolutional polynomial affect the formation of minimum-weight codewords in cosets under reverse precoding?
• RQ3 What is the performance gain in block error rate when using selective reverse PAC coding with sphere decoding compared to standard polar codes?
• RQ4 Does the proposed method achieve significant SNR gains in high-rate and high-SNR regimes?
• RQ5 Can the error coefficient be reduced without compromising the minimum distance, enabling practical sphere decoding of PAC codes?

Key findings

• The proposed SR-PAC coding reduces the error coefficient (A_w_min) by up to 92.5% compared to standard polar codes, particularly for high-rate codes like (64,50) and (128,110).
• For the (64,50) code, the error coefficient drops from 944 (polar) to 70 (SR-PAC(10)), resulting in a 0.2 dB SNR gain in low SNR and higher gains in high SNR.
• For the (128,110) code, the error coefficient is reduced from 4099 to 99, achieving up to 0.6 dB SNR gain under sphere decoding.
• The method maintains the minimum distance of the original polar code (e.g., w_min = 16 for (64,14) and (128,110)) by ensuring coset leaders have sufficient weight (w(g_i) = 16).
• Sphere decoding with SR-PAC outperforms SCL decoding with list size L=16 in high-SNR regimes for high-rate codes, indicating near-ML performance with lower complexity.
• The performance gain is most pronounced in high-rate and high-SNR regimes, where the error coefficient has the greatest impact on BLER.